The affine-bundle conjecture for p-adic Coxeter orbits

Let $X_{c,b}^{\caG_{\bfx}^{\rm ad}}$ be the affine Deligne–Lusztig variety associated with the Coxeter element $c$, the element $b$, and the adjoint parahoric group scheme $\caG_{\bfx}^{\rm ad}$. Let $\bA^\infty$ denote an infinite-dimensional affine space, and let the corresponding classical Deligne–Lusztig variety be attached to the reductive quotient of the special fiber of $\caG_{\bfx}$. The affine-bundle conjecture. $X_{c,b}^{\caG_{\bfx}^{\rm ad}}$ is an $\bA^\infty$-bundle over that classical Deligne–Lusztig variety. This gives a geometric description of the p-adic Coxeter orbit in terms of a classical Deligne–Lusztig variety; the supplied context does not state whether the claim has been proved or remains open.